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Complete left-invariant affine structures on nilpotent Lie groups. (English) Zbl 0591.53045

Which simply connected Lie groups admit a complete left-invariant affine structure, or equivalently which Lie groups act simply transitively on \({\mathbb{R}}^ n\) as affine transformations, is an important open question in the study of affine manifolds. We study some basic properties of such Lie groups through the left-symmetric product on its Lie algebra defined by the connection.
The classification problem of such structures is known for dimensions less than 4. The case of dimension 4 when the group is nilpotent is carried out in this paper. As an application of this classification one can easily determine all the possible left-invariant complete flat pseudo-Riemannian structures on the simply connected nilpotent Lie groups of dimension \(\leq 4.\)
This classification especially reveals that there are two isomorphism classes of complete left-invariant flat structures such that they do not contain any pure translations when they are viewed as simple transitive actions on \({\mathbb{R}}^ n\) by affine transformations.

MSC:

53C30 Differential geometry of homogeneous manifolds
22E25 Nilpotent and solvable Lie groups
53B05 Linear and affine connections
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